Closed-Form Solution Lagrange Multipliers in Worst-Case Performance Optimization Beamforming

This paper presents a robust adaptive rectangular antenna array beamforming using fixed diagonal loading (FDL) and optimal diagonal loading (ODL) techniques. Robustness of beamformer is important for practical implementation because of different uncertainties created in beamformer such as uncertainty in direction-of-arrival (DOA), imperfect antenna array calibration, near-far field effect, mutual coupling and other mismatches and modelling errors. Among different uncertainties DOA estimation error causes the most erroneous result. Numerical simulation results show that the robust technique improves the rectangular array performance, therefore improves the signal-to-noise plus interference ratio (SINR) of the beamformer. Results also show that the optimal diagonal loading (ODL) technique gives better performance than fixed diagonal loading (FDL) technique in terms of SINR.

Diagonal loading is one of the most widely used and effective methods to improve the robustness of both adaptive beamformers and Direction of Arrival (DOA) estimation due to the involvement of the sensor received covariance matrix. In addition, subspace-based DOA estimation techniques rely on multiple snapshots to achieve high estimation accuracies. This paper presents the study of a modified diagonally loaded sample covariance matrix for accurate DOA estimation in adverse scenarios. The proposed and novel technique deciphers poor DOA estimation in a low SNR environment by computationally changing the received sample covariance matrix. Our method is computationally simple as it does not require peak searching and does not depend on the coherency of the signal. The efficacy of the proposed method is examined via computer simulation for various sensor array sizes and the number of snapshot samples. Based on our numerical simulation results, our proposed method generally outperforms most state-of-the-art DOA estimators. In a finite number of snapshots and a single signal source, our proposed method performs 9.5% better than the state-of-the-art DOA estimation technique, 2.8% in multiple signal sources, and 8.5% in a single snapshot, single signal source environment of gained DOA estimation performance.

Robust adaptive beamforming is a key issue in array applications where there exist uncertainties about the steering vector of interest. Diagonal loading is one of the most popular techniques to improve robustness. Recently, worst-case approaches which consist of protecting the array's response in an ellipsoid centered around the nominal steering vector have been proposed. They amount to generalized (i.e. non necessarily diagonal) loading of the covariance matrix. In this paper, we present a theoretical analysis of the signal to interference plus noise ratio (SINR) for this class of robust beamformers, in the presence of random steering vector errors. A closed-form expression for the SINR is derived which is shown to accurately predict the SINR obtained in simulations. This theoretical formula is valid for any loading matrix. It provides insights into the influence of the loading matrix and can serve as a helpful guide to select it. Finally, the analysis enables us to predict the level of uncertainties up to which robust beamformers are effective and then depart from the optimal SINR.

The error in the look direction estimate can cause the mismatch between the actual and presumed steering vector. With this imprecise knowledge, the mainbeam is shifted leading to the self-null phenomenon of the desired signal and resulting in the performance degradation. In order to be robust against direction of arrival (DOA) mismatch, an adaptive diagonal loading technique is proposed and applied to the minimum variance distortionless response (MVDR) beamformer and the covariance matrix taper (CMT) null broadening beamforming. The adaptation is terminated by convergence of an indicator for the updated presumed DOA. The simulation results in comparison between a fixed diagonal loading and the proposed diagonal loading technique exhibits enhanced robustness and increased output signal to interference plus noise ratio (SINR).

If the desired signal is present in training snapshots, the adaptive array performance is known to be quite sensitive even to slight mismatches between the presumed and actual signal steering vectors. Such mismatches can occur as a result of environmental nonstationarities, look direction errors, imperfect array calibration or distorted antenna shape, as well as distortions caused by medium inhomogeneities, near-far mismatch, source spreading, and local scattering. The similar type of performance degradation can occur when the signal steering vector is known exactly but the training sample size is small. In this paper, we develop a new approach to robust adaptive beamforming in the presence of an arbitrary unknown signal steering vector mismatch. Our approach is based on the optimization of worst-case performance using Second-Order Cone (SOC) programming. The adaptive beamformer proposed is shown to have a substantially improved robustness as compared to existing algorithms and enjoy simple implementation.

Spherical harmonic decomposition facilitates decomposing the sound pressure at different microphones into independent functions of frequency, azimuth and elevation of the source and microphone locations. This decomposition facilitates the extraction of two sets of features containing different information about elevation and azimuth of the source for the direction of arrival (DOA) estimation. These features can be given as input to a learning approach for the estimation of azimuth and elevation separately. This approach aims at breaking down the problem of DOA estimation into azimuth and elevation estimation separately. An advantage of this is the reduction in computational complexity when compared with the joint DOA estimation. This facilitates a straightforward extension of this approach to denser DOA search grids. The contribution of this paper is threefold. First, we propose spherical harmonic magnitude and phase features and discuss the information present in these features regarding the azimuth and elevation of the source. Second, we propose the convolutional neural network architectures for DOA estimation. Third, we analyse the training, run-time computational complexities and propose to extend the DOA estimation approach to dense DOA search grid rather than restricting to a sparse DOA search grid. The performance of conventional DOA estimation approaches degrades in case of a noisy and reverberant environment. Several advancements to the existing DOA estimation approaches have been recently proposed. However, to the best of the authors’ knowledge, learning approaches to DOA estimation with dense DOA search grids with few frames in the context of spherical arrays have not been proposed. Performance evaluation is carried out using simulated as well as real datasets. The proposed approach is also evaluated on LOCATA dataset in the context of a moving source. The results are motivating enough to consider the application of the proposed method in practical scenarios.

Robust adaptive beamforming is a key issue in array applications where there exist uncertainties about the steering vector of interest. Diagonal loading is one of the most popular techniques to improve robustness. In this paper, we present a theoretical analysis of the signal-to-interference-plus-noise ratio (SINR) for the class of beamformers based on generalized (i.e., not necessarily diagonal) loading of the covariance matrix in the presence of random steering vector errors. A closed-form expression for the SINR is derived that is shown to accurately predict the SINR obtained in simulations. This theoretical formula is valid for any loading matrix. It provides insights into the influence of the loading matrix and can serve as a helpful guide to select it. Finally, the analysis enables us to predict the level of uncertainties up to which robust beamformers are effective and then depart from the optimal SINR.

Joint estimation of direction-of-departure (DOD) and direction-of-arrival (DOA) in bistatic multiple-input-multiple-output (MIMO) radar is a powerful technique for target localization. However, in the scenario of large arrays in which the numbers of array elements and observations are on the same order of magnitude, the sample covariance matrix cannot directly replace the statistical covariance matrix. In this paper, we propose two robust DOA and DOD estimation algorithms based on improved 2D-Capon estimators for MIMO radar with large arrays. The sample covariance matrix of the array received signals is modified by diagonal loading technique, and then the eigenvalues and eigenvectors of the diagonally loaded covariance matrix are accurately estimated utilizing Stieltjes transform and random matrix theory (RMT). Finally, two improved 2D-Capon estimators are derived to obtain robust performance of DOD and DOA estimation. The simulation shows that the improved algorithms can estimate angles more accurately than the traditional 2D-Capon algorithm in large dimension regime.

Direction of arrival (DOA) estimation is a fundamental problem in signal processing. The super-resolution spatial spectrum estimation algorithm based on feature decomposition, such as multiple signal classification (MUSIC) and estimating signal parameter via rotational invariance techniques (ESPRIT), has a good performance in high signal-to-interference plus noise ratio (SINR) environment and single interference. However, strong interferences will make DOA estimation error. And the number of angles that can be estimated by these methods is limited by the freedom of the antenna. The existence of multiple interferences will affect accuracy of DOA estimation. To address this problem, in this paper, we propose a novel dual-channel synchronization architecture and a Piloted-Aided DOA estimation method. We improve the matching performance by dual channel synchronization. A new matched filter is designed by using the idea of filter truncation, which reduces the computational resource cost. And a new DOA estimation method is proposed, which can estimate the angle of target signal in the environment of low SINR and multiple interferences. Experiments were performed to verify the performance of the proposed design. Results show that this architecture can greatly reduce the computational complexity. And the method successfully gets the direction of target signal in low SINR environment.

In this paper, novel robust adaptive beamformers are proposed with constraints on array magnitude response. With the transformation from the array output power and the magnitude response to linear functions of the autocorrelation sequence of the array weight, the optimization of an adaptive beamformer, which is often described as a quadratic optimization problem in conventional beamforming methods, is then reformulated as a linear programming (LP) problem. Unlike conventional robust beamformers, the proposed method is able to flexibly control the robust response region with specified beamwidth and response ripple. In practice, an array has many imperfections besides steering direction error. In order to make the adaptive beamformer robust against all kinds of imperfections, worst-case optimization is exploited to reconstruct the robust beamformer. By minimizing array output power with the existence of the worst-case array imperfections, the robust beamforming can be expressed as a second-order cone programming (SOCP) problem. The resultant beamformer possesses superior robustness against arbitrary array imperfections. With the proposed methods, a large robust response region and a high signal-to-interference-plus-noise ratio (SINR) enhancement can be achieved readily. Simple implementation, flexible performance control, as well as significant SINR enhancement, support the practicability of the proposed methods.

A robust linearly constrained minimum variance (LCMV) beamforming using directional and null constraints with a diagonal loading technique is discussed in this paper. The diagonal loading adds robustness against the direction of arrival mismatch and provides satisfactory output power and signal to interference plus noise ratio (SINR) in the mismatch region without broadening the beam. This paper first compares the proposed robust LCMV method with the conventional LCMV method in the presence of direction of arrival mismatch. Then its performance is compared with the LCMV using existed fixed diagonal loading method. The proposed robust method raises the output power by 1.88 dB and output SINR by 0.58 dB at 3 degree disparity angle compared to the LCMV with fixed diagonal loading technique. Simulations are performed in MATLAB to analyze the performance of the proposed method.

The main contribution of this thesis is to study the signal processing issues in MIMO radar and propose novel algorithms for improving the MIMO radar system. In the first part of this thesis, we focus on the MIMO radar receiver algorithms. We first study the robustness of the beamformer used in MIMO radar receiver. It is known that the adaptive beamformer is very sensitive to the DOA (direction-of-arrival) mismatch. In MIMO radar, the aperture of the virtual array can be much larger than the physical receiving array in the SIMO radar. This makes the performance of the beamformer more sensitive to the DOA errors in the MIMO radar case. In this thesis, we propose an adaptive beamformer that is robust against the DOA mismatch. This method imposes constraints such that the magnitude responses of two angles exceed unity. Then a diagonal loading method is used to force the magnitude responses at the arrival angles between these two angles to exceed unity. Therefore the proposed method can always force the gains at a desired interval of angles to exceed a constant level while suppressing the interferences and noise. A closed form solution to the proposed minimization problem is introduced, and the diagonal loading factor can be computed systematically by a proposed algorithm. Numerical examples show that this method has an excellent SINR (signal to noise-plus-interference ratio) performance and a complexity comparable to the standard adaptive beamformer. We also study the space-time adaptive processing (STAP) for MIMO radar systems. With a slight modification, STAP methods developed originally for the single-input multiple-output (SIMO) radar (phased array radar) can also be used in MIMO radar. However, in the MIMO radar, the rank of the jammer-and-clutter subspace becomes very large, especially the jammer subspace. It affects both the complexity and the convergence of the STAP algorithm. In this thesis, we explore the clutter space and its rank in the MIMO radar. By using the geometry of the problem rather than data, the clutter subspace can be represented using prolate spheroidal wave functions (PSWF). Using this representation, a new STAP algorithm is developed. It computes the clutter space using the PSWF and utilizes the block diagonal property of the jammer covariance matrix. Because of fully utilizing the geometry and the structure of the covariance matrix, the method has very good SINR performance and low computational complexity. The second half of the thesis focuses on the transmitted waveform design for MIMO radar systems. We first study the ambiguity function of the MIMO radar and the corresponding waveform design methods. In traditional (SIMO) radars, the ambiguity function of the transmitted pulse characterizes the compromise between range and Doppler resolutions. It is a major tool for studying and analyzing radar signals. The idea of ambiguity function has recently been extended to the case of MIMO radar. In this thesis, we derive several mathematical properties of the MIMO radar ambiguity function. These properties provide some insights into the MIMO radar waveform design. We also propose a new algorithm for designing the orthogonal frequency-hopping waveforms. This algorithm reduces the sidelobes in the corresponding MIMO radar ambiguity function and makes the energy of the ambiguity function spread evenly in the range and angular dimensions. Therefore the resolution of the MIMO radar system can be improved. In addition to designing the waveform for increasing the system resolution, we also consider the joint optimization of waveforms and receiving filters in the MIMO radar for the case of extended target in clutter. An extended target can be viewed as a collection of infinite number of point targets. The reflected waveform from a point target is just a delayed and scaled version of the transmitted waveform. However, the reflected waveform from an extended target is a convolved version of the transmitted waveform with a target spreading function. A novel iterative algorithm is proposed to optimize the waveforms and receiving filters such that the detection performance can be maximized. The corresponding iterative algorithms are also developed for the case where only the statistics or the uncertainty set of the target impulse response is available. These algorithms guarantee that the SINR performance improves in each iteration step. The numerical results show that the proposed iterative algorithms converge faster and also have significant better SINR performances than previously reported algorithms. (Abstract shortened by UMI.)

Significant effort has gone into designing robust adaptive beamforming algorithms to improve robustness against uncertainties in array manifold. These uncertainties may be caused by uncertainty in direction-of-arrival (DOA), imperfect array calibration, near-far effect, mutual coupling, and other mismatch and modeling errors. A diagonal loading technique is obligatory to fulfil the uncertainty constraint where the diagonal loading level is amended to satisfy the constrained value. The major drawback of diagonal loading techniques is that it is not clear how to get the optimum value of diagonal loading level based on the recognized level of uncertainty constraint. In this paper, an alternative realization of the robust adaptive linearly constrained minimum variance beamforming with ellipsoidal uncertainty constraint on the steering vector is developed. The diagonal loading technique is integrated into the adaptive update schemes by means of optimum variable loading technique which provides loading-on-demand mechanism rather than fixed, continuous or ad hoc loading. We additionally enrich the proposed robust adaptive beamformers by imposing a cooperative quadratic constraint on the weight vector norm to overcome noise enhancement at low SNR. Several numerical simulations with DOA mismatch, moving jamming, and mutual coupling are carried out to explore the performance of the proposed schemes and compare their performance with other traditional and robust beamformers

To solve the problem of the direction of arrival (DOA) and Doppler mismatch errors between the actual and presumed signal steering vectors in airborne radar spacetime adaptive processing (STAP), a robust bi-iterative STAP algorithm based on second-order cone programming (SOCP) is proposed. Explicit models of uncertainties in both temporal and spatial signal steering vectors are considered. It is shown that the robust STAP problem can be solved by minimizing a convex quadratic cost function based on the optimization of worst-case performance when full degrees of freedom (DOFs) are used. The quadratic cost function is further reformulated into a bi-quadratic cost function by adopting a separable form for the weight vector and the optimum solution of the new cost function can be efficiently obtained by combining bi-iterative algorithm (BIA) with SOCP. The proposed method can achieve the robust STAP performance while significantly decreasing the training sample requirement and the computational complexity, which is more valuable for practical engineering application. Simulation results are provided to demonstrate the effectiveness of the proposed robust STAP method.

We propose a Taylor expansion multiple signal classification (TE MUSIC) method for joint direction of departure (DOD) and direction of arrival (DOA) estimation in a bistatic multiple-input multiple-output (MIMO) array. First, using a Taylor expansion of the steering vector, a two-dimensional (2D) search in the conventional MUSIC method for MIMO arrays is reduced to a two-step one-dimensional (ID) search in the proposed TE MUSIC method. Second, DOAs of the targets can be achieved via Lagrange multiplier by a ID search. Finally, substituting the DOA estimates into the 2D MUSIC spectrum function, DODs of the targets are obtained by another ID search. Thus, the DOD and DOA estimates can be automatically paired. The performance of the proposed method is better than that of the MIMO ESPRIT method, and is similar to that of the 2D MUSIC method. Furthermore, due to the ID search, the TE MUSIC method avoids the high computational complexity of the 2D MUSIC method. Simulation results are presented to show the effectiveness of the proposed method.